256 Prof. F. Y. Edge worth on the Application of 



not now instantaneous, and that the reciprocal couple 

 consists of different genera as well as classes. But it can 

 now no longer be taken for granted that the differential 

 factor of the frequency-content relating to any species of 

 couple — of the form 



AQi AQ 2 . . . AU AY AQi AQ 2 . . .Aq l Aq 2 . . .Au Av A£i Aq 2 



(/. c. p. 267) — (where U, Y, u,v are velocities of the mass- 

 centres and the other symbols relate to generalized co- 

 ordinates) retains its value unchanged after the change 

 which is signified by affecting each of the above symbols 

 with a dash. To secure the equality of the differential 

 factors before and after an encounter, recourse must be had 

 to the theorem of Liouville. 



In general, in order that the distribution of velocities 

 should be stable, it is necessary that it should obey the 

 normal law of frequency in two forms : one form (proper to 

 each genus) in which the coefficients of (the squares and 

 products of) the velocities are functions of the co-ordinates, 

 and another form (pertaining to the medley as a whole) in 

 which the coefficients do not involve co-ordinates. 



2. Proof that the normal distribution is sufficient. — To show 

 that this distribution is sufficient, as well as necessary, 

 recourse must be had to a priori probability (I. c. p. 257) : the 

 presumptions which attach to the hypothesis of random 

 distribution. Of this kind is the generally admitted pro- 

 position that the frequency of a class is not altered by 

 changing the signs of all the components of velocity of 

 momentum (I. c. p. 266). Thus in the simple case first 

 "considered where the section (V , Y ; u, v)s passed by collision 

 into (u, v 1 ; XT', V')s; the latter section is equal in content to 

 the section designated by the same velocities with signs 

 reversed, in our notation ( — "LP, — V ; — u[, —v')s ; the 

 capital letters now preceding as the section is positive (con- 

 silient). By our third argument (and simple dynamics) the 

 last written section passes by collision into (— u, —v; 

 — U, — V)s; which is a priori equal to (IP, V; u, v)s. 

 Thus the content of: the section first considered is the same 

 at the end of the interval t as it was at the beginning. The 

 contents of this section and of every other section thus 

 remain constantly the same if the normal law of frequency 

 holds good. This proof may be transferred to the general 

 case of several degrees of freedom more directly than 

 appeared before (I. c. p. 265). 



3. Correlation. — There is of course interdependence 

 between molecules engaged in an encounter. Thus in the 



